9.7: Geometry of Hyperbolas

A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation is

Each branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equations

A rectangle centered at the origin with dimensions 2a by 2b helps locate asymptotes by extending its diagonals. The eccentricity, given by

where ​, always exceeds one and quantifies the hyperbola’s openness. The foci lie at (±c,0). Direct simplification of equations enables plotting and analysis. Hyperbolic mirrors in optical systems, such as Cassegrain telescopes, focus light precisely to enhance image clarity. The geometry of hyperbolas finds applications in astronomy, engineering, and navigation, where precise directional properties are essential.

A hyperbola consists of two open curves called branches. If P is a point on the curve, then the distances from P to the two foci are measured, and the absolute difference of these distances is constant. No matter where P is chosen on either branch, this difference remains the same.

The standard equation for a hyperbola centered at the origin and opening along the x-axis is x squared over a squared minus y squared over b squared equals one.

Each branch of a hyperbola approaches two diagonal lines, called asymptotes, that guide the curve to infinity.

Two times a gives the distance between the vertices along the transverse axis, while two times b defines the conjugate axis length.

The equations of the asymptotes depend on both a and b, which determine the slopes of the diagonal lines of the central rectangle. Using the point–slope formula, the equations of the asymptote lines can then be written.

Hyperbolas also appear in optical instruments. In astronomy, the Cassegrain telescope uses a parabolic primary and a hyperbolic secondary mirror. The primary focuses incoming parallel rays, and the secondary, sharing one focus with the primary, reflects them toward its second focus through a hole in the primary to form an image.